| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 54·5-s + 36·6-s + 49·7-s − 64·8-s + 81·9-s + 216·10-s + 564·11-s − 144·12-s + 476·13-s − 196·14-s + 486·15-s + 256·16-s − 1.30e3·17-s − 324·18-s + 14·19-s − 864·20-s − 441·21-s − 2.25e3·22-s + 529·23-s + 576·24-s − 209·25-s − 1.90e3·26-s − 729·27-s + 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.965·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.683·10-s + 1.40·11-s − 0.288·12-s + 0.781·13-s − 0.267·14-s + 0.557·15-s + 1/4·16-s − 1.09·17-s − 0.235·18-s + 0.00889·19-s − 0.482·20-s − 0.218·21-s − 0.993·22-s + 0.208·23-s + 0.204·24-s − 0.0668·25-s − 0.552·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| 23 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 54 T + p^{5} T^{2} \) |
| 11 | \( 1 - 564 T + p^{5} T^{2} \) |
| 13 | \( 1 - 476 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1308 T + p^{5} T^{2} \) |
| 19 | \( 1 - 14 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6702 T + p^{5} T^{2} \) |
| 31 | \( 1 + 10258 T + p^{5} T^{2} \) |
| 37 | \( 1 - 14078 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2826 T + p^{5} T^{2} \) |
| 43 | \( 1 - 14216 T + p^{5} T^{2} \) |
| 47 | \( 1 + 9990 T + p^{5} T^{2} \) |
| 53 | \( 1 - 16698 T + p^{5} T^{2} \) |
| 59 | \( 1 - 19044 T + p^{5} T^{2} \) |
| 61 | \( 1 - 5678 T + p^{5} T^{2} \) |
| 67 | \( 1 + 39736 T + p^{5} T^{2} \) |
| 71 | \( 1 - 60108 T + p^{5} T^{2} \) |
| 73 | \( 1 + 7714 T + p^{5} T^{2} \) |
| 79 | \( 1 - 106472 T + p^{5} T^{2} \) |
| 83 | \( 1 + 7662 T + p^{5} T^{2} \) |
| 89 | \( 1 + 76536 T + p^{5} T^{2} \) |
| 97 | \( 1 + 19048 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.991323309486170526426405992931, −8.009519057128156191549241420301, −7.24322693916013387409876331047, −6.48507886507496221618110106853, −5.57224838337369285654630775865, −4.20507948091677939966079811735, −3.70312765697564789537030882182, −2.01370320555062652858954455067, −1.01704524024465910934758982293, 0,
1.01704524024465910934758982293, 2.01370320555062652858954455067, 3.70312765697564789537030882182, 4.20507948091677939966079811735, 5.57224838337369285654630775865, 6.48507886507496221618110106853, 7.24322693916013387409876331047, 8.009519057128156191549241420301, 8.991323309486170526426405992931
